Problem: Find the least common multiple $(\text{LCM})$ of $25k^4+200k^3+400k^2$ and $15k^3+90k^2+120k$. You can give your answer in its factored form.
Solution: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $25k^4+200k^3+400k^2$ can be factored as ${(5)}{(5)(k)}{(k)(k+4)}{(k+4)}$ by factoring out a $25k^2$ and using the perfect square pattern. $15k^3+90k^2+120k$ can be factored as ${(3)}{(5)(k)(k+4)}{(k+2)}$ by factoring out a $15k$ and using the sum-product pattern. We can see that: Both polynomials share the factors ${(5)(k)(k+4)}$ Only the first polynomial has the factors ${(5)(k)(k+4)}$ Only the second polynomial has the factors ${(3)(k+2)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(5)(k)(k+4)}{(5)(k)(k+4)}{(3)(k+2)}\\\\ &=75(k^2)(k+4)^2(k+2)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $75(k^2)(k+4)^2(k+2)$.